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Table of Contents
the mode 53 range and interquartile range 53 outliers 53 standard deviation n 54 effect of adding scores to a set 54 effect of altering all the scores in a set 54 suitability of statistical displays 55 statistical misrepresentation in displays 56 comparison of like sets of data 56 The Normal Distribution 56 the normal distribution 56 standard normal distribution 57 probabilities and normal distributions 57 empirical values/empirical rule 58 z-scores 59
Functions
Working with Functions 1
definitions and notation ● Relation: in the form of a set of ordered pairs ● Function : relation in which no 2 ordered pairs have the same first component ○ For every value of x there is only one value of y ○ Can be specified by a rule (equation) in terms of x and y ○ The output y depends on the input x ○ x is the independent variable and y is the dependent variable
○ y is a function of x → 𝑦 = 𝑓(𝑥)
● All functions are relations but all relations are not functions ● Piecewise/piecemeal: function defined by different rules for different domains domain and range ● Domain : set of all possible x values for which the function is defined ○ May be restricted ● Range : set of all values of y obtained over the domain of the function
Note: denominator cannot equal 0 and a number below a square root sign cannot be negative interval notation ● [square brackets]: inclusion of an endpoint ● (curved parentheses): exclusion of an endpoint ○ [𝑎, 𝑏]→ interval extends from a to b, with a and b included ○ (𝑎, 𝑏)→ interval extends from a to b, with a and b excluded ○ [𝑎, 𝑏)→ interval extends from a to b, with a included and b excluded ○ (𝑎, 𝑏]→ interval extends from a to b, with a excluded and b included ○ (− ∞, ∞)→ interval includes the set of all real numbers R ● Interval notation can be used to represent the domain and range of a function the graph of a function ● Vertical line test: if at least one vertical line crosses a graph more than once, then the graph is not a function ● The domain of a function determines the width of its graph ● The range of a function determines the height of its graph ● The graph of 𝑦 = 𝑓(𝑥)has x-intercepts where y = 0, and y-intercepts where x = 0 ● Graph is increasing when y increases as x increases ● Graph is decreasing when y decreases as x increases ● The values at which the graph 𝑦 = 𝑓(𝑥) crosses the x-axis are the solutions of the equation 𝑓(𝑥) = 0
● 𝑂𝑑𝑑 ×÷ 𝐸𝑣𝑒𝑛 = 𝑁𝑒𝑖𝑡ℎ𝑒𝑟 ● 𝐸𝑣𝑒𝑛 ×÷ 𝑂𝑑𝑑 = 𝑁𝑒𝑖𝑡ℎ𝑒𝑟 four types of relations One-to-one function: for every value of x there is only one value of y and for every value of y there is only one value of x ● Passes both vertical and horizontal line tests Many-to-one function: more than one value of x has the same value of y ● Passes the vertical line test, fails the horizontal line test One-to-many relation: at least one value of x has more than one value of y ● Fails the vertical line test, passes the horizontal line test Many-to-many relation: more than one value of x has more than one value of y ● Fails both the vertical and horizontal line tests
composite functions ● Composite function: involves the output of one function rule becoming the input of another function rule ○ 𝑓(𝑔(𝑥))and 𝑔(𝑓(𝑥))represent composite functions ● 𝑓(𝑔(𝑥)) ≠ 𝑔(𝑓(𝑥))→ are different results ● The notation: ○ (𝑓 ◦ 𝑔)(𝑥)is equivalent to 𝑓(𝑔(𝑥)) ○ (𝑔 ◦ 𝑓)(𝑥)is equivalent to 𝑔(𝑓(𝑥)) ● (𝑓 ◦ 𝑓)(𝑥) =𝑓(𝑓(𝑥)) = 𝑓 2 (𝑥) ● When determining the domain and range 𝑓(𝑔(𝑥))given 𝑓(𝑥)and 𝑔(𝑥), it is a better option to find the equation of 𝑓(𝑔(𝑥))and determine its domain and range gradient of linear functions 𝐺𝑅𝐴𝐷𝐼𝐸𝑁𝑇 = 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑅𝑖𝑠𝑒 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑅𝑢𝑛 𝑚 = 𝑦 2 −𝑦 1 𝑥 2 −𝑥 2 𝑚 = 𝑡𝑎𝑛Θ equations of straight lines ● 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0→ general form of the equation (a is positive) ● 𝑦 = 𝑚𝑥 + 𝑏→ oblique line with a gradient of m and y-intercept of b ● 𝑦 = 𝑚𝑥→ oblique line through the origin with a gradient of m ● 𝑥 = 𝑎→ vertical line which crosses the x-axis at a (contains only the x variable) ● 𝑦 = 𝑏→ horizontal line which crosses the y-axis at b (contains only the y variable) ● 𝑦 − 𝑦 → line with a gradient of m that passes through 1 = 𝑚(𝑥 − 𝑥 1 ) the point (𝑥 1 , 𝑦 1 )
- Substitute the pair of values into the equation in order to find k
- Rewrite the equation with the correct value of k linear functions as models of physical phenomena ● Increasing linear model equation: ○ 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = (𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 × ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦) + 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒 ○ e.g. 𝐹 = 10ℎ + 20 ● Decreasing linear model equation: ○ 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 − (𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 × ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑞𝑢𝑎 ○ e.g. 𝐹 = 200 − 5𝑤 ● Features: ○ Gradient is a rate of change which compares the vertical quantity to one unit of the horizontal quantity ○ The vertical intercept has a significant meaning in each model ○ The independent variable is shown on the horizontal axis ○ The dependent variable is shown on the vertical axis simultaneous equations ● The solution to a pair of simultaneous equations graphed on the same set of axes is the coordinates of the point of intersection ● The point of intersection of cost and income graphs is called a break-even point ○ Here, production = equal income ○ This point can be established algebraically by equating the two equations and then solving to find the first (x) coordinate of the point general forms of quadratic equations ● 𝑎𝑥 → expression 2
- 𝑏𝑥 + 𝑐 ● 𝑎𝑥 → equation 2
- 𝑏𝑥 + 𝑐 = 0 ● 𝑦 = 𝑎𝑥 → graph 2
- 𝑏𝑥 + 𝑐
solving quadratic equations
- Factorise → equate factors to 0 → solve
- Complete perfect square
- Quadratic formula 𝑥 = −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 solving quadratic inequalities ● Critical points method: ○ Solve as an equation ○ Plot points on number line ○ Test regions ● Parabola method: ○ Sketch parabola showing x-intercepts ○ Determine where graph lies above or below or on the x-axis graphs of quadratic functions The graph of 𝑦 = 𝑎𝑥 is a parabola, which... 2
- 𝑏𝑥 + 𝑐 ● Is concave up if a > 0 or concave down if a < 0 ● Cuts the y-axis at c ● Has the y-axis as its axis of symmetry if b = 0 ● Has x-intercepts where 𝑎𝑥 2
- 𝑏𝑥 + 𝑐 = 0 ● Has an axis of symmetry given by 𝑥 =− 𝑏 2𝑎 ● Has a maximum/minimum value when 𝑥 =− 𝑏 2𝑎 ● Will turn on the x-axis if 𝑎𝑥 is a perfect square 2
- 𝑏𝑥 + 𝑐 the vertex ● When the equation of a quadratic has 2 unequal roots, these roots are the x-intercepts of the parabola
→ solve 𝑎𝑥 2
- 𝑏𝑥 + 𝑐 = 𝑚𝑥 + 𝑏 ● To show there are: ○ 2 points → show ∆> 0 ○ 1 point → show ∆= 0 ○ 0 points → show ∆< 0 modelling quadratic functions ● Problems involving direct linear variation may require a quadratic function to describe the relationship ○ This occurs when one variable is directly proportionate to the square of another ● The graph of this type of quadratic relationship is a parabola (equations: 𝑦 = 𝑘𝑥 where k is a constant) 2 ● When the independent variable x is multiplied by a factor, the dependent variable is multiplied by the square of the factor ● e.g: ○ when x is doubled, y is multiplied by 4 ○ when x is halved, y is multiplied by 1 4 ○ when x is multiplied by 5, y is multiplied by 25 ○ when x is divided by 3, y is divided by 9 modelling cubic functions ● Problems involving direct linear variation may require a cubic function to describe the relationship ○ This occurs when one variable is directly proportionate to the cube of another ● The graph of this type of quadratic relationship is a cubic curve (equations: 𝑦 = 𝑘𝑥 where k is a constant) 3 ● When the independent variable x is multiplied by a factor, the dependent variable is multiplied by the cube of the factor the cubic curve General form: 𝑦 = 𝑎𝑥 3
- 𝑏𝑥 2
- 𝑐𝑥 + 𝑑
● Shape: ○ Decreases to the left and increases to the right when a > 0 ○ Increases to the left and decreases to the right when a < 0 ● Domain: all real x ● Range: all real y ● 𝑦 = 𝑎𝑥 → inflects at the origin; odd function 3 ● 𝑦 = 𝑎𝑥 → inflects on the y-axis at c 3
- 𝑐 ● 𝑦 = (𝑥 − 𝑎) → inflects on the x-axis at a 3 ● 𝑦 = (𝑥 − 𝑎) → inflects at (a, c) 3
- 𝑐 ● 𝑦 = (𝑥 − 𝑎)(𝑥 − 𝑏)(𝑥 − 𝑐) → crosses the x-axis at a, b and c
Working with Functions 2
polynomial definitions 𝑃(𝑥) = 𝑎 𝑛 𝑥 𝑛
- 𝑎 𝑛− 𝑥 𝑛− +... + 𝑎 1 𝑥 + 𝑎 0 ● n = 0, 1, 2, ... ● 𝑎 are real numbers 0 , 𝑎 1 , 𝑎 2 ,..., 𝑎 𝑛 ● 𝑎 is the leading term 𝑛 𝑥 𝑛 ● Has a degree of n ○ Polynomials of degree 2 are called quadratic polynomials ○ Polynomials of degree 3 are called cubic polynomials ○ Polynomials of degree 4 are called quartic polynomials ● Is monic if 𝑎 𝑛 = 1 ● 𝑎 is the constant term 0 ● A zero of the polynomial expression P(x) is a root or solution of the polynomial equation P(x) = 0 and an x-intercept of the graph y = P(x) graphs of polynomial functions ● A polynomial of degree n cannot cut the x-axis more than n times ● A polynomial of odd degree must cut the x-axis at least once ● A polynomial of even degree may not cut the x-axis at all ● A polynomial graph must have at least one turning point between two
hyperbolas ● Functions of the form 𝑓(𝑥) = , where k is a constant, represent inverse 𝑘 𝑥 variation ● The graph of 𝑦 = is hyperbolic in shape with the following features: 𝑘 𝑥 ○ Discontinuous with branches in the 1st and 3rd quadrants when k > 0 and branches in the 2nd and 4th quadrants when k < 0 ○ x and y axes are asymptotes ○ Has odd symmetry ○ Domain: all real x, x ≠ 0 ○ Range: all real y, y ≠ 0 ● 𝑦 = is a vertical translation of by c units, with asymptotes 𝑘 𝑥
- 𝑐 𝑦 = 𝑘 𝑥 of x = 0 and y = c ● 𝑦 = is a horizontal translation of by b units, with asymptotes 𝑘 𝑥−𝑏 𝑦 = 𝑘 𝑥 of x = b and y = 0 absolute value graphs Linear absolute value graphs ● Linear graphs involving absolute value have a V shape, either upwards or downwards ● The function of the linear graph is 𝑦 = | |𝑥 ● If the sign of the function is: ○ Positive, the V points upwards ○ Negative, the V points downwards ● 𝑦 = | | + 𝑐𝑥 is a vertical translation of the graph by c units ● 𝑦 = |𝑥 − 𝑏 |is a horizontal translation of the graph by b units ● If the whole function rule is inside the absolute value signs, then the vertex (corner) of the graph will be on the x-axis ○ When this is the case finding the x and y intercepts will help sketch the graph ● Another method for sketching linear graphs in which the whole function is inside the absolute value sign, is to:
- Sketch inside the graph ignoring the absolute value sins
- Reflect the section below the x-axis above the x-axis
- Erase the section below the x-axis ● The graph 𝑦 = | |𝑥has a:
○ Domain of all real x ○ Range of 𝑦 ≥ 0 ● To sketch absolute value graphs with a restricted domain, plot the endpoints and any axial intercepts Absolute value graphs ● To draw any absolute value graph in the form 𝑦 = |𝑓(𝑥) |, you draw the graph of 𝑦 = 𝑓(𝑥)and then reflect any sections that are below the x-axis above the x-axis circles ● Circles are not functions ● 𝑥 → centre (0, 0), radius r units 2
- 𝑦 2 = 𝑟 2 ● (𝑥 − 𝑎) → centre (a, b), radius r units 2
- (𝑦 − 𝑏) 2 = 𝑟 2 ● Equations of circles in the form 𝑥 are 2
- 𝑦 2
- 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 transformed into the form (𝑥 − 𝑎) by completing the 2
- (𝑦 − 𝑏) 2 = 𝑟 2 square semi-circles upper semi-circle ● 𝑦 = 𝑟 2 − 𝑥 2 ○ Centre (0, 0) ○ Radius r ○ Even function ○ Domain: − 𝑟 ≤ 𝑥 ≤ 𝑟 ○ Range: 0 ≤ 𝑦 ≤ 𝑟 lower semi-circle ● 𝑦 =− 𝑟 2 − 𝑥 2 ○ Centre (0, 0) ○ Radius r ○ Even function ○ Domain: − 𝑟 ≤ 𝑥 ≤ 𝑟 ○ Range: − 𝑟 ≤ 𝑦 ≤ 0
𝑎𝑠𝑖𝑛𝑥 + 𝑏𝑐𝑜𝑠𝑥 = 𝑐 → 𝑅𝑠𝑖𝑛(𝑥 + α) = 𝑐 𝑎𝑠𝑖𝑛𝑥 − 𝑏𝑐𝑜𝑠𝑥 = 𝑐 → 𝑅𝑠𝑖𝑛(𝑥 − α) = 𝑐 𝑎𝑐𝑜𝑠𝑥 + 𝑏𝑠𝑖𝑛𝑥 = 𝑐 → 𝑅𝑐𝑜𝑠(𝑥 − α) = 𝑐 𝑎𝑐𝑜𝑠𝑥 − 𝑏𝑠𝑖𝑛𝑥 = 𝑐 → 𝑅𝑐𝑜𝑠(𝑥 + α) = 𝑐 quadratic trigonometric equations ● If the equation is quadratic you factorise, and then form two separate equations by equating the factors to zero ● The two equations are then solved independently ● e.g. 𝑠𝑖𝑛 2 𝑥 − 𝑠𝑖𝑛𝑥 = 0 𝑠𝑖𝑛𝑥(𝑠𝑖𝑛𝑥 − 1) = 0 𝑠𝑖𝑛𝑥 = 0 𝑠𝑖𝑛𝑥 = 1 x = 0, 90, 180, 360 trigonometric identities quotient identities 𝑡𝑎𝑛θ = 𝑠𝑖𝑛θ 𝑐𝑜𝑠θ 𝑐𝑜𝑡θ = 𝑐𝑜𝑠θ 𝑠𝑖𝑛θ reciprocal identities 𝑐𝑜𝑡θ = 1 𝑡𝑎𝑛θ 𝑐𝑜𝑠𝑒𝑐θ = 1 𝑠𝑖𝑛θ 𝑠𝑒𝑐θ = 1 𝑐𝑜𝑠θ Pythagorean identities 𝑠𝑖𝑛 2 θ + 𝑐𝑜𝑠 2 θ = 1 𝑡𝑎𝑛 2 θ + 1 = 𝑠𝑒𝑐 2 θ 1 + 𝑐𝑜𝑡 2 θ = 𝑐𝑜𝑠𝑒𝑐 2 θ double angle formulas 𝑠𝑖𝑛2θ = 2𝑠𝑖𝑛θ𝑐𝑜𝑠θ 𝑐𝑜𝑠2θ = 𝑐𝑜𝑠 2 θ − 𝑠𝑖𝑛 2 = 2𝑐𝑜𝑠 2 θ − 1 = 1 − 2𝑠𝑖𝑛 2 θ 𝑡𝑎𝑛2θ = 2𝑡𝑎𝑛θ 1−𝑡𝑎𝑛 2 θ sum identities addition/subtraction formulas 𝑠𝑖𝑛(𝑎 ± 𝑏) = 𝑠𝑖𝑛(𝑎)𝑐𝑜𝑠(𝑏) ± 𝑐𝑜𝑠(𝑎)𝑠 𝑐𝑜𝑠(𝑎 ± 𝑏) = 𝑐𝑜𝑠(𝑎)𝑐𝑜𝑠(𝑏) ∓ 𝑠𝑖𝑛(𝑎)𝑠 𝑡𝑎𝑛(𝑎 ± 𝑏) = 𝑡𝑎𝑛(𝑎)±𝑡𝑎𝑛(𝑏) 1∓𝑡𝑎𝑛(𝑎)𝑡𝑎𝑛(𝑏) co-function identities 𝑐𝑜𝑠( π 2 − θ) = 𝑠𝑖𝑛θ 𝑠𝑖𝑛( π 2 − θ) = 𝑐𝑜𝑠θ 𝑐𝑜𝑡( π 2 − θ) = 𝑡𝑎𝑛θ 𝑡𝑎𝑛( π 2 − θ) = 𝑐𝑜𝑡θ even-odd identities 𝑠𝑖𝑛(− θ) =− 𝑠𝑖𝑛θ 𝑐𝑜𝑠(− θ) = 𝑐𝑜𝑠θ 𝑡𝑎𝑛(− θ) =− 𝑡𝑎𝑛θ 𝑐𝑜𝑠𝑒𝑐(− θ) =− 𝑐𝑜𝑠𝑒𝑐 𝑠𝑒𝑐(− θ) = 𝑠𝑒𝑐θ 𝑐𝑜𝑡(− θ) =− 𝑐𝑜𝑡θ half-angle formulas 𝑠𝑖𝑛( θ 2 ) =± 1−𝑐𝑜𝑠θ 2
𝑐𝑜𝑠𝑒𝑐( π 2 − θ) = 𝑠𝑒𝑐θ 𝑠𝑒𝑐( π 2 − θ) = 𝑐𝑜𝑠𝑒𝑐θ 𝑐𝑜𝑠( θ 2 ) =± 1+𝑐𝑜𝑠θ 2 𝑡𝑎𝑛( θ 2 ) = 1−𝑐𝑜𝑠θ 𝑠𝑖𝑛θ = 𝑠𝑖𝑛θ 1+𝑐𝑜𝑠θ =± 1−𝑐𝑜𝑠θ 2 sum-to-product formulas 𝑠𝑖𝑛(𝑎) + 𝑠𝑖𝑛(𝑏) = 2𝑠𝑖𝑛( 𝑎+𝑏 2 )𝑐𝑜𝑠( 𝑎−𝑏 2 𝑠𝑖𝑛(𝑎) − 𝑠𝑖𝑛(𝑏) = 2𝑐𝑜𝑠( 𝑎+𝑏 2 )𝑠𝑖𝑛( 𝑎−𝑏 2 𝑐𝑜𝑠(𝑎) + 𝑐𝑜𝑠(𝑏) = 2𝑐𝑜𝑠( 𝑎+𝑏 2 )𝑐𝑜𝑠( 𝑎−𝑏 2 𝑐𝑜𝑠(𝑎) − 𝑐𝑜𝑠(𝑏) =− 2𝑠𝑖𝑛( 𝑎+𝑏 2 )𝑠𝑖𝑛( 𝑎− 2 product-to-sum formulas 𝑠𝑖𝑛(𝑎)𝑠𝑖𝑛(𝑏) = 1 2 [𝑐𝑜𝑠(𝑎 − 𝑏) − 𝑐𝑜𝑠(𝑎 + 𝑏 𝑐𝑜𝑠(𝑎)𝑐𝑜𝑠(𝑏) = 1 2 [𝑐𝑜𝑠(𝑎 + 𝑏) + 𝑐𝑜𝑠(𝑎 − 𝑏 𝑠𝑖𝑛(𝑎)𝑐𝑜𝑠(𝑏) = 1 2 [𝑠𝑖𝑛(𝑎 + 𝑏) + 𝑠𝑖𝑛(𝑎 − 𝑏 𝑐𝑜𝑠(𝑎)𝑠𝑖𝑛(𝑏) = 1 2 [𝑠𝑖𝑛(𝑎 + 𝑏) − 𝑠𝑖𝑛(𝑎 − 𝑏 tips when solving trig equations ● Division of both sides of an equation by a trig ratio may remove some of the solutions ○ Tip: factorise then solve ● Substitution of one ratio with its fractional equivalent may remove some of the solutions ○ Tip: factorise then solve OR equate the denominator to zero to find the other possible solutions using the t-formulae ● Substitution using the t-formulae can be applied when solving equations of the form: ○ 𝑎𝑠𝑖𝑛𝑥 ± 𝑏𝑐𝑜𝑠𝑥 = 𝑐 ○ 𝑎𝑐𝑜𝑠𝑥 ± 𝑏𝑠𝑖𝑛𝑥 = 𝑐 𝑠𝑖𝑛(𝑥) = 2𝑡 1+𝑡 (^2) 𝑐𝑜𝑠(𝑥) = 1−𝑡 2 1+𝑡 2 ● Where 𝑡 = 𝑡𝑎𝑛( 𝑥 2 ) ● However, when this method is used the possible solution of x = πis lost, as
